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AN  INVESTIGATION 

OF  THE  SECOND  LAW  OF 

THERMODYNAMICS 


By 

JACOB  T.  WAINWRIGHT 


CHE 


CHICAGO 


o 


Printed  June  1913 


PREFACE 

The  feasibility  of  a  self -actuating  refrigerating  machine,  that 
is,  "the  perpetual  motion  of  the  second  kind,"  has  ceased  to  be  a 
question  universally  identified  as  "absurd,"  "necessarily  impos- 
sible," "an  indication  of  ignorance,"  et  cetera.  Furthermore, 
throughout  the  scientific  world,  at  the  present  time,  a  rapidly 
increasing  multitude  is  becoming  seriously  interested  in  the  out- 
come of  this  question  and  an  early  positive  settlement  will  afford 
general  satisfaction. 

This  brief  contribution  to  the  science  of  thermodynamics  is  an 
investigation  pursued  along  the  strictest  lines  of  philosophy  and 
reasoning  with  the  view  of  determining  what  particular  positive 
knowledge  is  now  lacking  and  must  be  acquired  in  order  positively 
to  settle  this  question. 

The  investigation  discloses  that  all  knowledge  involved  is 
known  positively,  excepting  exact  determination  of  the  value  or 
measure  of  the  variations  of  the  " disgregation  of  expansion"  and 
the  two  kinds  of  "specific  heat"  which  result  by  reason  of  changes 
in  physical  state  as  determined  by  changed  conditions  of  pressure, 
density,  and  temperature.  However,  positive  knowledge  pertain- 
ing to  this  relation,  although  somewhat  meager  and  obtained  from 
sporadic  experimentation,  is  vouched  for  by  some  of  the  most 
eminent  investigators  of  the  present  day,  and  indicates  that  "per- 
petual motion  of  the  second  kind"  is  feasible. 

The  matter  of  co-ordinating  these  six  manifestations  as  func- 
tional variables,  instead  of  only  the  usual  three,  constitutes  a 
much-needed  reform  in  the  science  of  thermodynamics,  and  con- 
sequently means  a  more  complex  science.  However,  since  Nature 
is  not  simple  but  is  infinitely  complex,  all  advancement  in  science 
necessarily  means  progression  from  simple  crudeness  to  complex 
refinement. 

Therefore,  this  paper  is  further  purposed  as  a  plea  for  experi- 
mental research  to  establish  extended  positive  knowledge  regarding 

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this  functional  relation;   and  thereby  indirectly  settle  the  question 
at  issue,  positively  and  finally. 

To  savants  with  desire  to  pursue  this  subject  further  and  in 
different  manner,  the  author  would  mention  several  of  his  papers 
which  were  read  at  the  meetings  of  the  American  Association  for 
the  Advancement  of  Science,  beginning  in  the  year  1903,  and  are 
available  from  the  reference  shelves  of:  the  National  Library  of 
Congress,  Washington,  B.C. ;  the  Library  of  the  Franklin  Institute, 
Philadelphia;  the  Astor-Tilden-Lennox  Library,  New  York  City; 
the  Crerar  Library,  Chicago;  also,  some  communications  published 
in  The  Engineer  (London):  see  issues  of  June  21,  1912,  p.  658; 
July  26,  1912,  p.  90;  September  6,  1912,  p.  259. 

J.  T.  W. 


THE  SECOND  LAW  OF  THERMODYNAMICS 

The  " principle  of  cause  and  effect"  is  our  fundamental  analyti- 
cal comprehension  of  a  happening,  and  consequently  is  our  funda- 
mental analytical  criterion  for  "Fact."  This  principle  means  that 
cause  and  effect  are  true  equivalents. 

Phenomena  constitute  the  only  source  from  which  positive 
knowledge  is  derived;  consequently,  as  relates  to  physical  science, 
all  truly  established  facts  or  principles  must  be  derived,  either  di- 
rectly or  indirectly,  from  manifestations  of  Nature  revealed  either 
directly  or  by  the  aid  of  experimentation.  Consequently,  as  relates 
to  physical  science,  a  postulation  necessarily  imposes  on  Nature 
an  exacting  requirement  which  must  be  upheld  by  the  physical 
properties  of  matter  in  order  to  uphold  the  postulation  as  a  truth. 

Indirect  derivation  is  properly  effected  by  co-ordinating  a 
plurality  of  more  directly  derived  facts.  However,  it  must  ever 
be  kept  in  mind  that  the  scope  of  an  indirectly  derived  fact  or 
principle  cannot  be  considered  as  established  beyond  the  limitations 
imposed  by  the  established  facts  comprised  in  the  particular  co- 
ordination from  which  it  is  derived.  Each  co-ordinated  fact 
necessarily  contributes  its  quota  to  the  formulation  of  the  resultant 
derived  fact  or  principle,  and  such  distinct  contribution  is  properly 
identified  as  the  significance  of  that  particular  contributing  fact; 
furthermore,  this  means  that  the  significance  of  a  fact  or  principle 
is  not  necessarily  a  fixed  contribution  for  all  co-ordinations,  but  is 
dependent  upon  the  other  facts  or  principles  which  are  comprised 
in  the  particular  co-ordination  considered. 

It  will  be  perceived  that  when  a  postulation  is  comprised  in  the 
co-ordination,  a  positive  fact  or  principle  cannot  be  derived  there- 
from, but  necessarily  a  derived  postulation  results.  Also,  subse- 
quent discovery  of  additional  pertinent  facts  or  principles  which 
should  have  been  included  in  the  co-ordination  necessarily  rele- 
gates the  principle  which  was  derived  from  such  incomplete  source 
to  its  proper  limitations  as  hereinbefore  described,  and  necessitates 

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the  derivation  of  a  new  principle  which  must  be  in  accord  with  a 
revised  co-ordination. 

Since  advancement  in  science  eventually  discloses  new  facts  and 
principles  which  are  pertinent  and  should  have  been  included  in  the 
co-ordination  from  which  accepted  principles  were  derived,  it 
necessarily  results  that  promulgation  of  a  principle  as  a  generaliza- 
tion devoid  of  limitations  is  naturally  abhorrent  to  all  scientists 
possessed  of  true  acumen. 

From  what  precedes,  it  will  be  perceived  that  in  order  to 
investigate  the  truth  of  a  principle  and  locate  an  underlying  falsity 
if  it  exists,  when  direct  verification  by  experimentation  is  inexpe- 
dient, it  becomes  necessary: 

First:  To  determine  what  positively  known  facts  or  principles 
are  pertinent  in  the  matter. 

Second:  To  determine  if  the  principle  results  from  a  co- 
ordination of  a  complete  list  of  such  facts,  or  only  from  a  partial 
list  of  same,  or  only  from  a  list  in  which  postulation  is  involved. 

Third:  To  determine  what  requirement  is  imposed  by  postu- 
lation, and  to  determine  if  it  is  upheld  by  the  properties  of  matter 
involved;  and  further  to  determine  to  what  extent  or  limitations 
the  properties  of  matter  can  uphold  such  postulation. 

The  science  of  thermodynamics,  in  its  present-day  state,  is 
founded  on  two  fundamental  underlying  principles;  one  is  an  axiom, 
the  other  is  a  bald  postulation. 

The  " first"  fundamental  underlying  principle  or  law  is  com- 
monly known  as  the  "first  law  of  thermodynamics,"  and  is  merely 
a  specific  formulation  of  that  which  in  a  more  generic  formulation 
is  known  as  "the  principle  of  conservation  of  energy,"  which  again 
in  a  more  generic  formulation  is  known  as  "  Newton's  principle  of 
action  and  reaction,"  which  again  in  a  more  generic  formulation  is 
known  as  the  "  principle  of  equivalence  in  cause  and  effect. "  Since, 
as  was  hereinbefore  explained,  the  generic  formulation  of  this  princi- 
ple is  our  fundamental  analytical  criterion  for  "Fact,"  necessarily 
it  is  truly  an  axiom. 

The  "second"  fundamental  underlying  principle  is  simply  the 
denial  of  the  so-called  chimera  known  as  "the  perpetual  motion  of 
the  second  kind."  In  the  following  simple  manner,  Carnot  demon- 
strated that  this  "denial"  necessarily  is  the  resultant  of  the  co- 

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ordination  of  "the  principle  of  conservation  of  energy,"  with  "the 
materialistic  theory  of  heat": 

"An  unlimited  creation  of  motive  power,  without  consumption 
either  of  caloric  or  of  any  other  agent  whatever,  constitutes  per- 
petual motion.  Such  a  creation  is  entirely  contrary  to  ideas  now 
accepted,  to  the  laws  of  mechanics  and  of  sound  physics.  It  is 
inadmissible." 

As  a  mere  philosopheme,  it  may  here  be  remarked  that,  since 
this  demonstration  shows  in  what  manner  this  "denial"  is  accu- 
rately derived  from  a  co-ordination  of  two  principles,  destruction  of 
one  of  these  principles  necessarily  means  destruction  of  the  derived 
principle.  Consequently,  the  replacement  of  the  theory  of  inde- 
structible material  heat  by  the  conversional  theory  necessarily 
causes  this  "denial"  to  become  untenable. 

Carnot  subsequently  arrived  at  this  same  conclusion,  and  is 
explained  farther  on. 

Also,  it  may  here  be  mentioned  as  an  interesting  fact  that  the 
late  Henri  Poincare,  throughout  his  writings,  frequently  expressed 
dissatisfaction  with  the  "second  law,"  and  in  one  instance  as  follows: 

"The  second  law  of  thermodynamics  does  not  seem  to  be  com- 
patible with  the  first  law." 

It  seems  that  the  manner  by  which  he  arrived  at  this  con- 
clusion has  never  been  disclosed. 

In  a  manner  which  does  not  involve  the  nature  of  heat,  Carnot 
showed  that  a  cyclic  acting  machine  to  effect  "perpetual  motion 
of  the  second  kind"  means  a  combination  of  a  heat  motor  with  a 
heat  pump,  in  which  the  thermodynamic  agent  of  the  motor  must 
be  more  efficient  than  that  of  the  pump,  by  an  amount  sufficient 
to  overcome  the  mechanical  inefficiencies  of  the  entire  apparatus, 
in  addition  to  the  useful  work  derived;  and  that,  when  the  appa- 
ratus is  assumed  to  operate  in  a  perfect  manner,  that  is,  free  from  all 
manner  of  irreversibleness,  the  assumption  of  imperativeness  neces- 
sarily imposes  the  following  principle: 

"The  motive  power  of  heat  is  independent  of  the  agents  em- 
ployed to  realize  it;  its  quantity  is  fixed  solely  by  the  tempera- 
ture of  the  bodies  between  which  is  effected,  finally,  the  transfer  of 
the  caloric." 


This  is  Carnot's  formulation  of  what  is  now  known  as  the 
"  second  law  of  thermodynamics,"  and  has  never  been  improved 
upon  by  the  many  ambitious  writers  who  are  identified  with  its 
various  formulations  extant.  It  will  be  perceived  that,  although 
the  derivation  of  the  second-kind  perpetual  motion  " denial" 
involves  the  nature  of  heat,  this  so-called  "second  law"  is  derived 
from  this  "denial"  in  a  manner  which  does  not  further  involve  a 
consideration  of  the  nature  of  heat  and  consequently  agrees  with 
the  present-day  state  of  the  science.  Furthermore,  it  will  be 
perceived  that  it  is  not  a  formulation  of  a  negation,  neither  is  it 
verbose,  but  concisely  formulates  the  limitations  which  apply 
to  the  possibilities  of  a  heat  engine,  as  determined  by  accepting 
,  this  "denial"  and  co-ordinating  same  with  "the  principle  of  con- 
servation of  energy." 

Also,  in  a  manner  which  does  not  involve  the  nature  of  heat,  Carnot 
followed  this  matter  up  and  demonstrated  that  the  specific  mani- 
festation of  Nature  which  this  "second  law"  imposes  is  that,  in  a 
reversible  cycle  in  which  the  applied  heat  is  transferred  at  one 
temperature  and  the  discharge  of  heat  is  effected  at  one  other 
temperature,  the  average  rate  of  conversion  of  a  given  quantity  of 
applied  heat  must  be  the  same  for  all  agents,  for  the  same  range  of 
temperature;  and,  furthermore,  that  this  requirement  must  hold 
true  for  all  ranges  of  pressure,  or  of  density,  in  which  the  cycle  may 
be  operated.  It  results  as  a  mathematical  deduction  from  this 
principle  that  the  efficiency  function,  that  is,  Carnot's  function, 
must  be  the  same  for  all  substances  at  the  same  temperature. 
Since  the  manner  of  deducing  this  principle  from  the  "second  law" 
does  not  involve  the  nature  of  heat,  it  necessarily  agrees  with  the 
present-day  state  of  the  science,  and  is  the  crux  which  has  never 
been  satisfactorily  upheld,  either  directly  or  indirectly,  by  reliable 
knowledge  derived  from  experimentation. 

Rate  of  conversion  means:  "Amount  of  external  work  produced 
by  a  unit  of  temperature-drop";  and  as  applies  to  such  reversible 
cyclic  conversion,  it  is  commonly  known  as  "the  thermodynamic 
function";  and  when  this  function  corresponds  to  a  unit  measure 
of  applied  heat,  it  is  known  as  "Carnot's  function." 

However,  Carnot  subsequently  repudiated  the  materialistic 
theory  of  heat,  the  second-kind  perpetual  motion  "denial"  which 

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he  had  derived  from  same,  the  so-called  "'second  law"  which  he 
had  derived  from  this  "denial,"  and  necessarily  all  of  the  principles 
which  he  had  derived  from  this  "  second  law,"  unequivocally,  and 
by  the  following  precise  statement: 

"When  a  hypothesis  no  longer  suffices  to  explain  phenomena,  it 
should  be  abandoned.     This  is  the  case  with  the  hypothesis  which 

regards  caloric  as  matter,  as  a  subtle  fluid But  it  would  be 

difficult  to  explain  why,  in  the  development  of  motive  power  by  heat, 
a  cold  body  is  necessary;  why,  in  consuming  the  heat  of  a  warm  body, 
motion  cannot  be  produced." 

Thus,  in  this  brief  observation  in  regard  to  a  cold  body,  Carnot 
not  only  states  that  it  would  be  difficult  to  explain  why  "perpetual 
motion  of  the  second  kind"  is  not  possible,  but  at  the  same  time 
he  explains  what  it  means. 

Also,  in  this  manner,  he  has  certainly  established  his  identity 
as  the  original  pioneer  in  the  movement  to  eliminate  the  pernicious 
falsity  which  at  the  present  time  underlies  the  entire  science. 

In  order  to  explain  this  seeming  anomaly,  it  may  here  be  proper 
briefly  to  repeat  some  history  which  has  already  been  published 
elsewhere,  and  explain  that: 

"  A  sparse  edition  of  Carnot's  now  famous  treatise  on  the  Motive 
Power  of  Heat  was  published  in  the  year  1824,  that  he  died  suddenly 
in  the  year  1832.  However,  in  the  intervening  time,  by  reason  of 
his  changed  belief  regarding  the  nature  of  heat,  he  recorded  some 
notes  and  memoranda  apparently  with  the  view  to  publishing  a 
revised  treatise  and  also  to  prosecute  some  important  experimenta- 
tions. Fortunately,  his  original  manuscript  of  these  notes  and  also 
of  his  published  treatise  were  preserved  in  the  archives  of  the  French 
Academy  of  Science  where  they  remained  in  oblivion  until  the  year 
1878  when  they  were  published  in  their  entirety." 

His  aforementioned  repudiation  of  the  "second  law,"  and  also 
his  formulation  of  what  is  now  known  as  the  "Joule-Mayer  principle 
of  conservation  of  energy"  were  comprised  in  these  notes. 

Professors  Rudolph  Clausius  and  William  Thomson  (Lord 
Kelvin)  were  chiefly  responsible  for  the  present-day  deplorable 
state  of  the  particular  branch  of  the  science  of  thermodynamics 
which  relates  to  the  motive  power  of  heat,  particularly  regarding 
the  "second  law"  question.  Both  had  acquired  preponderant 

9 


reputation  as  mathematicians,  and  respectively  announced  that 
they  had  established  the  "second  law "  as  true  by  reconciling  experi- 
mentally derived  facts  with  mathematical  deduction.  This 
announcement  from  such  eminent  source,  unfortunately,  seems  to 
have  diverted  censorial  scrutiny,  and  was  swallowed  as  gospel 
truth  by  almost  the  entire  scientific  world. 

Clausius  arrived  at  this  conclusion  by  properly  assuming  that, 
sufficient  for  all  mundane  affairs,  some  one  of  the  so-called  per- 
manent gases  is  amenable  to  the  "Boyle-Gay  Lussac  law,"  and  that 
consequently  the  location  of  a  so-called  absolute-zero  on  a  perfect- 
gas  thermometer  is  accurately  expressed  by  the  reciprocal  of  such 
fluid's  "coefficient  of  expansion."  Upon  this  assumption  he  accu- 
rately deduced  that,  for  such  fluid,  the  rate  of  reversible  cyclic 
conversion  of  a  given  quantity  of  applied  transferred  heat  is  neces- 
sarily a  measure  or  value  which  is  expressed  by  ratioing  the  quan- 
tity of  this  transferred  heat  by  the  so-called  absolute-temperature 
at  which  it  is  transferred.  Furthermore,  he  showed  that,  in  order 
to  uphold  the  "second  law"  it  becomes  necessary  that  this  formulation 
must  also  hold  true  for  all  substances. 

Transferred  heat,  ratioed  by  so-called  absolute-temperature 
corresponding  to  such  transfer,  is  a  mathematical  conception  which 
he  calls  "entropy";  and  from  the  principle  just  enunciated,  he 
accurately  derived  the  following  principle: 

"The  second  law  means  that,  in  a  reversible  cycle,  the  aggre- 
gate of  positive  entropy  must  be  equal  to  the  aggregate  of  negative 
entropy." 

In  that  manner,  Clausius  accurately  brought  the  question  to  the 
following  simple  positive  issue  having  but  one  alternative: 

Either  abandon  the  "second  law";  or  abandon  experimentally 
derived  values  for  " disgregation  of  expansion/'  and  the  two  kinds  of 
"specific-heat,"  when  a  co-ordination  of  same  to  determine  the  rate 
of  reversible  cyclic  conversion  does  not  uphold  "entropy"  as  the  true 
measure  of  such  rate,  for  all  regions  of  pressure,  density,  and  tempera- 
ture, and  also  for  all  substances. 

When  Clausius  brought  the  question  to  this  abrupt  conclu- 
sion, Kelvin  had  expended  much  effort  during  several  years  in  for- 
mulating efficiency  tables  for  steam  and  other  heat  engines,  based 
on  the  experimentations  of  Regnault  and  other  investigators. 


These  tables  cover  merely  a  meager  range  of  pressure,  but  never- 
theless disclose  a  very  considerable  discrepancy  from  the  require- 
ment imposed  by  the  Clausius  ultimatum.  However,  Kelvin 
finally  recognized  the  alternative  nature  to  which  the  question  had 
been  accurately  brought,  and  concluded  that  such  discrepancies 
necessarily  must  be  attributed  to  inaccuracies  of  experimentation. 
Both,  it  would  seem  by  mutual  assent,  then  abandoned  further 
consideration  and  discussion  of  the  experimental  phase  of  the 
question,  and  adopted  the  postulated  "second  law"  as  the  true 
alternative. 

This  certainly  does  not  appear  to  be  a  very  strict  reconciliation 
of  postulation  with  experimentation,  and  possibly  each  foresaw  in 
such  procedure  an  easy  road  to  much  fame:  for  one,  the  announce- 
ment to  the  scientific  world  of  a  finite  absolute-zero  on  Nature's 
scale  of  temperature;  for  the  other,  the  announcement  that  "the 
entropy  of  the  Universe  tends  toward  a  maximum,"  and  the  con- 
sequent significance  regarding  ultimate  death  of  Nature. 

Kelvin's  absolute-zero  doctrine  has  already  been  destroyed  by 
reason  of  advancement  in  the  art  of  refrigeration  whereby  lique- 
faction of  the  so-called  permanent  gases  became  a  reality,  co- 
ordinated with  the  fact  that  "specific-heat  decreases  rapidly  with 
increased  cold." 

Also,  already,  present-day  reliable  determinations  for  "specific 
heat"  and  " disgregation  of  expansion,"  as  derived  from  experi- 
mentations by  eminent  investigators,  disclose  reliable  facts  which 
cannot  be  reconciled  with  the  aforementioned  deductions  which 
Clausius  accurately  derived  from  the  "second  law." 


wny  3 


THE  NEW 
THERMODYNAMICS 


AN  ADDENDUM  TO  PAPER  HAVING  THE  TITLE: 

"AN  INVESTIGATION  OF  THE  SECOND 

LAW  OF  THERMODYNAMICS" 


JACOB  T.  WAINWRIGHT 


CHICAGO 


Second  Edition 
Printed  October  igi3 


THE  NEW  THERMODYNAMICS 

The  principal  purpose  of  this  brief  contribution  to  the  science  of 
thermodynamics  is  to  present  a  simple  and  positive  mathematical 
refutation  of  the  " second  law,"  and  incidentally  to  determine  what 
particular  peculiarities  must  be  possessed  by  a  substance  in  order 
that  it  can  be  amenable  to  the  Clausius  formulation  for  efficiency. 
Another  purpose  is  to  present  what  seems  to  be  a  simple  and 
desirable  foundation  upon  which  to  rebuild  the  entire  science  of 
" motive  power  of  heat." 

"Specific  heat,"  reckoned  as  variable  instead  of  as  constant,  is 
the  particular  fundamental  feature  which  is  peculiar  to  this  new 
departure,  and  consequently  introduces  a  further  rate  of  change  or 
additional  differentiation  into  the  mathematics  of  the  science. 

Furthermore,  postulation  is  taboo.  Consequently,  by  reason  of 
the  fact  that  " specific  heat"  is  always  a  variable  function  to  a 
greater  or  less  degree  for  all  substances,  the  conception  of  a  finite 
thermometric  location  for  absolute  zero  of  temperature  will  not  be 
tolerated.  Therefore,  "temperature"  must  be  reckoned  in  the 
relative  sense.  However,  volume  and  pressure  may  be  reckoned  in 
the  absolute  sense. 

The  necessity  of  holding  to  this  qualification  becomes  apparent 
from  a  consideration  of  the  fact  that  the  idea  of  "absolute  tem- 
perature," as  commonly  conceived  at  the  present  day,  was  derived 
from  a  mathematical  conception  which  is  functionally  affected  by 
variation  of  "specific  heat." 

It  does  not  seem  necessary  to  explain  .why  an  investigation  of  the 
"motive  power  of  heat,"  for  the  purpose  of  this  particular  research, 
simply  reduces  to  a  study  of  the  efficiency  of  a  Carnot  cycle. 

It  is  purposed  to  refute  the  "second  law"  by  an  accurate  and 
systematic  mathematical  determination  of  the  theoretical  efficiency 
which  is  obtainable  from  various  kinds  and  conditions  of  working 
media,  when  operating  in  a  Carnot  cycle  which  is  identically  the 
same  for  all  in  regard  to  extent  of  temperature  range  and  also  in 
regard  to  the  temperature  location  of  such  range,  but  without  any 

3 


other  restrictions.  Therefore,  this  manner  of  procedure  enables  a 
comparison  of  such  results  to  show  that:  For  the  same  range  of 
temperature,  " possible  maximum  efficiency"  is  not  the  same  for  all 
kinds  and  conditions  of  working  media,  and  consequently  consti- 
tutes a  positive  refutation  of  the  " second  law,"  by  refuting  a 
so-called  principle  which  is  supposed  to  uphold  the  " second  law" 
in  a  manner  that  is  understood  by  all  students  of  thermodynamics. 

Applying  the  calculus  of  ratios  or  rates,  as  conceived  by  Newton 
and  Leibnitz,  and  keeping  in  mind  the  usual  pressure-volume 
diagram  of  a  Carnot  cycle  in  which  the  vertical  ordinates  represent 
pressure  and  the  horizontal  ordinates  represent  volume,  it  is  per- 
ceived that  such  diagram  by  means  of  a  series  of  vertical  lines  may 
be  subdivided  into  differentiations  of  volume,  and  in  like  manner  by 
means  of  horizontal  lines  may  be  subdivided  into  differentiations 
of  pressure. 

This  manner  of  differentiating  means  that,   throughout  this 

mathematical  treatment  of  the  subject,  ~r  represents  the  rate  of 
change  of  pressure  condition  corresponding  to  unchanged  volume, 
and  -j-  in  like  manner  represents  the  rate  of  change  of  volume 

condition  corresponding  to  unchanged  pressure.  Therefore,  the 
fundamental  formulation  from  \\hich  to  derive  the  evaluation  for 
the  external  work  of  the  entire  cycle  may  be  written: 

d*W  =  dpXdv,  (i) 

in  which  W  represents  the  external  work  produced  by  the  entire 

cycle. 

Consequently,  the  derived  integral  equation  which  represents 

the  evaluation  of  the  external  work  produced  by  the  entire  cycle  may 

be  formulated  thus: 

Expansion  adiabatic 

dpXdv 

Compression  adiabatic 


in  us: 

H2~Hl=    I         I 
J  U    J 


in  which  ta  represents  the  temperature  at  which  the  applied  heat 
H2  is  transferred  to  the  medium,  and  ^  represents  the  lesser  tem- 
perature at  which  the  medium  discharges  the  heat  HIf 


The  "rate"  of  such  cyclic  conversion,  with  respect  to  temperature 
drop,  necessarily  means  the  portion  of  such  diagrammatic  area  which 
represents  an  imaginary  Carnot  cycle  comprised  between  the  same 
adiabatic  limits  but  corresponding  to  a  unit  of  temperature  drop. 
Consequently,  the  work  produced  by  the  entire  cycle  represents  the 
aggregate  or  summation  of  a  succession  of  such  smaller  cycles 
which  respectively  correspond  to  each  unit  of  the  temperature  drop 
of  the  entire  cycle,  and  each  intermediate  cycle  comprised  in  such 
series  is  supposed  to  receive  its  supply  of  applied  transferred  heat 
from  the  one  which  next  precedes,  and  to  transfer  its  discharged 
heat  into  the  one  which  next  follows.  This  conception  originated 
from  Carnot. 

The  next  procedure  is  to  derive  from  equation  (2)  the  evaluation 
of  the  work  which  is  represented  by  one  of  these  rate  cycles  located 
at  any  representative  temperature  /. 

By  differentiating  equation  (2)  with  respect  to  temperature, 
there  results: 

•Expansion  adiabatic 

dpXdv  (3) 

Compression  adiabatic. 


y-  f 

J 


/ 


By  ratioing  equation  (3)  in  a  manner  which  is  equivalent  to  an 
integration  for  temperature  limits  which  correspond  to  a  range  of 
temperature  restricted  to  a  unit  measure  of  extent,  there  results: 

Expansion  adiabatic 

(4*) 

Compression  adiabatic 
Expansion  adiabatic 

(4*) 
Compression  adiabatic 

which  is  the  basic  formulation  for  rate  of  cyclic  conversion  with 
regard  to  temperature  drop,  as  applies  to  a  Carnot  cycle.  This 
particular  evaluation  is  commonly  known  as  "the  thermodynamic 
function"  and  is  usually  designated  by  the  symbol  <£,  or  sometimes 
(<j>a—<t>b) ;  and  when  the  limitations  for  the  cyclic  limiting  adiabatics 

5 


are  restricted  so  as  to  correspond  to  an  initial  application  of  a  unit 
of  transferred  heat,  it  is  known  as  "the  efficiency  function,"  that  is, 
"Carnot's  function." 

However,  this  form  of  the  evaluation  is  not  well  adapted  to 

practical  use,  and  therefore  it  becomes  desirable  that  the  ratio  ~  , 

which  is  an  expression  that  represents  "constant  volume  change  of 
pressure"  corresponding  to  a  unit  change  of  temperature,  shall  be 


expressed  by  y^x^      ,  in  which  ^    is  the  symbol  for  the  "coeffi- 

cient of  pressure  change,"  that  is,  a  factor  which  is  always  a  function 
of  temperature,  and  sometimes  also  a  function  of  pressure  or  of 
volume,  and  which,  when  multiplied  by  the  corresponding  pressure, 
results  in  a  product  that  represents  such  particular  pressure  change. 

Also,  in  like  manner  it  becomes  desirable  that  the  ratio  -j  ,  which 

is  an  expression  that  represents  "constant  pressure  change  of  volume" 
corresponding  to  a  unit  change  of  temperature,  shall  be  expressed 


by  \^x^      ,  in  which  ^    is  the  symbol  for  the  "coefficient  of 

volume  change,"  that  is,  a  factor  which  is  always  a  function  of 
temperature,  and  sometimes  also  a  function  of  volume  or  of  pres- 
sure, and  which,  when  multiplied  by  the  corresponding  volume, 
results  in  a  product  that  represents  such  particular  volume  change; 
furthermore,  it  may  be  explained  that  this  particular  coefficient  is  a 
conception  which  is  not  new  to  the  science,  and  is  known  as  the 
coefficient  of  expansion. 

Therefore,  by  substitution  of  these  expressions,  equations  (40) 
and  (46)  become: 

Expansion  adiabatic 


/ 

/ 


Compression  adiabatic 
Expansion  adiabatic 


Compression  adiabatic 
6 


in  which  p  and  v  apply  only  to  the  involved  representative  isother- 
mal line;   also  I  ~  I  and  I  ~  J  respectively  may  be  a  function  of  /,  p, 


and  v. 

Since  "specific  heat"  is  a  concept  which  means:  quantity  of 
heat  required  to  effect  a  unit  change  of  tenperature  to  a  unit  of 
mass,  necessarily  it  results  that, 


and  in  like  manner 


in  which  (Kv)  represents  "specific  heat  for  constant  volume," 
(Kp)  represents  "specific  heat  for  constant  pressure,"  and  M  rep- 
resents the  mass  of  the  working  fluid  and  is  a  constant  measure  or 
value.  Therefore,  by  substituting  these  expressions,  equations 
(50)  and  (56)  become: 

Expansion  adiabatic 
dW 


J 
/ 


dt=  +  =  I      ^T 

Compression  adiabatic 

(Expansion  adiabatic 
&^Xdp  (6b) 

•p 

Compression  adiabatic 


in  which  v  and  p  apply  only  to  the  involved  representative  isother- 
mal line;  also  (Kv)  and  (Kp)  respectively  may  be  a  function  of  /, 
pj  and  v. 

These  four  equations  (50),  (56),  (6a),  and  (6b)  necessarily 
are  fundamental  or  generic  formulations.  Furthermore,  equations 
(6a)  and  (6b)  are  particularly  interesting  because  they  disclose  the 
subtle  significance  of  "specific  heat,"  as  it  affects  cyclic  efficiency, 
which  is  a  matter  that  was  not  known  or  properly  understood  in  the 
prior  state  of  the  science. 

Having  thus  formulated  the  true  generalization  for  "rate"  of 
Carnot  cyclic  conversion,  it  becomes  proper  to  continue  the  mathe- 

7 


matics  from  this  stage  merely  as  a  matter  of  progressive  specializa- 
tion from  generic  to  specific,  in  successive  steps-  corresponding 
respectively  to  the  evolution  of  an  involved  variable  into  a  constant, 
and  thereby  at  each  of  such  stages  separate  for  further  specializa- 
tion toward  the  ultimate  goal  of  physical  constants  as  exemplified 
by  a  "perfect  gas"  certain  corresponding  groups  of  substances  as 
determined  by  experimentally  derived  knowledge  regarding  their 
physical  properties. 

From  the  viewpoint  of  pure  mathematics,  eaeh  of  these  succes- 
sive eliminating  stages  constitutes  a  refutation  of  the  claim  that  the 
formulation  finally  derived  in  such  manner  and  applicable  only  to  a 
"perfect  gas"  is  a  true  generalization,  that  is,  holds  true  for  all 
kinds  and  conditions  of  matter.  This,  in  itself,  is  sufficient  to  end 
the  investigation,  then,  and  there.  However,  it  is  purposed  to 
progress  the  classification  to  the  ultimate  limit  of  specialization, 
and  compare  with  same  the  experimentally  measured  evaluations 
for  "rate  of  conversion"  corresponding  to  some  saturated  vapor 
which  has  been  investigated  experimentally,  is  typical  for  all 
saturated  vapors,  and  plainly  shows  the  typical  characteristics; 
and  thereby  disclose  discrepancies  which  are  imposed  by  Nature 
and  cannot  be  questioned. 

Therefore,  the  next  procedure  is  to  derive  from  equation  (50)  a 
less  generic  formulation  which  will  apply  only  to  a  substance  which 

maintains  a  constant  value  for  I  ^  I  during  the  entire  change  of 

volume  at  constant  temperature  /  between  the  limiting  adiabatics. 
Consequently,  such  derived  equation  necessarily  will  hold  true  for 
the  "saturated  vapor"  type  of  Carnot  cycle,  also  for  the  Carnot 
cycle  of  a  "perfect  gas,"  but  will  not  hold  true  for  an  "imperfect 
gas."  Consequently,  this  particular  step  of  gradation  causes 
equation  (50)  to  become: 

Expansion  adiabatic 

pXdv  (7) 

Compression  adiabatic 

in  which  p  applies  only  to  the  involved  representative  isothermal 
line;  and  I  ^  I  is  a  function  of  temperature  only,  but  not  neces- 


sarily  a  constant  function,  nor  necessarily  the  same  value  for  all 
substances. 

The  next  procedure  is  to  derive  from  this  equation  (7)  a  formu- 
lation which  again  is  less  generic,  and  will  apply  only  to  a  "perfect 
gas."  In  such  case,  the  Boyle-Gay  Lussac  law  imposes  the  feature 
that  " specific  heat"  is  constant  for  all  conditions  of  state;  con- 
sequently all  locations  on  an  isothermal  line  necessarily  correspond 
to  a  condition  of  unchanged  " intrinsic  energy,"  which  means 
absence  of  proclivity  to  divert  applied  heat  into  so-called  "latent 
energy";  therefore,  necessarily  /pXdv  now  represents  the  equiva- 
lent of  the  applied  transferred  heat  and  is  conceived  to  be  received 
at  constant  temperature  and  from  the'  next  preceding  cycle  and 
is  symbolized  as  h.  Therefore,  by  substituting  this  expression, 
equation  (7)  becomes: 

h ,  (8a) 

in  which  I  ^  I  is  not  a  constant,  but  must  be  evaluated  for  the  par- 
ticular isothermal  involved.  However,  the  Boyle-Gay  Lussac  law 
also  imposes  the  additional  feature  that: 

XpXv  =  Constant. 

Furthermore,  by  reason  of  acceptance  of  Dalton's  law,  it  is 
believed  that  this  particular  constant  is  common  to  all  "perfect 
gases"  having  a  common  value  for  pXv  at  a  common  temperature, 
and  may  be  derived  from  any  one  of  such  by  simple  experi- 
mentation; nevertheless,  this  particular  Dalton  principle,  as  a 
generalization  without  limitations,  is  somewhat  questionable; 
however,  this  question  does  not  affect  this  investigation. 


Also,  for  a  "perfect  gas"  I  ^    necessarily  has  the  same  evalua- 
tion as  I  £J .     Consequently,  equation  (80)  may  be  \\ritten: 


dW  = 

It  is  important  to  note  that  equation  (7)  marks  the  particular 
stage  in  the  process  of  successive  derivation  from  the  generic  to  the 

9 


specific,  where  parting  of  the  ways  occurs  for  the  " saturated  vapor" 
type  of  cycle,  and  the  " perfect  gas"  type.  This  matter  will  be 
commented  upon  farther  on. 

This  equation  (Sb)  is  the  famous  formulation  which  Clausius 
accurately  arrived  at  by  a  different  method  of  procedure.  His  pur- 
pose was  to  arrive  at  a  formulation  which  would  hold  true  for  a 
" perfect  gas."  He  was  not  interested  in  deductions  of  a  more 
generic  nature  because  he  did  not  intend  to  question  the  "  second 
law";  and  he  accurately  perceived  that  acceptance  of  that  law  neces- 
sarily means  that  an  accurately  derived  formulation  of  the  "thermo- 
dynamic function"  for  any  substance  must  also  hold  true  for  all 
substances,  and  that  a  " perfect  gas"  is  the  simplest  thermodynamic 
medium  from  a  mathematical  point  of  view  and  consequently  the 
most  convenient  to  investigate. 

Also,  Kelvin  was  not  eager  to  question  the  "second  law,"  but 
attempted  to  evaluate  "the  thermodynamic  function"  in  tabulated 
form  derived  from  known  properties  of  steam.  However,  he  finally 
accepted  the  Clausius  argument,  and  abandoned  his  experimentally 
derived  tables  because  they  showed  a  very  considerable  discrepancy  in 
comparison  with  evaluations  derived  by  the  Clausius  method  which 
does  not  necessitate  any  experimentation  other  than  one  determina- 
tion for  "coefficient  of  expansion"  for  some  one  "perfect  gas." 

Referring  to  equation  (7),  and  relating  to  working  media  which 
are  adapted  for  use  in  the  "saturated  vapor"  type  of  cycle  and  are 
amendable  to  this  equation,  Nature  has  not  imposed  a  value  for 

I  £-1  which  will  compensate  for  the  vagaries  of  disgregation  and 

uphold  a  common  "efficiency  function,"  and  thereby  refutes  the 
"second  la\\ ." 

Furthermore,  all  of  such  kind  of  substances  differ  from  a 
"perfect  gas"  in  a  manner  which  is  manifested  by  a  difference  in  the 

evaluation  of  I  ^  |X^Xft ;  and  since  this  expression  is  a  constant  for 

the  "perfect  gas,"  and  is  variable  for  these  other  substances,  it 
results  that  their  respective  evaluations  for  <£  will  depart  from  the 
"perfect  gas' '  <£  to  an  extent  which  is  a  function  of  pressure.  This 
necessarily  constitutes  an  additional  refutation  of  the  "second  law." 
Relating  to  the  "saturated  vapor"  type  of  Carnot  cycle,  the 


dW 
Clausius  conception  of  entropy  is  a  true  measure  of  the  initial  -JT- 

only  for  a  particular  temperature  having  a  scale  location  which  is 
different  for  different  substances;  whereas,  at  other  temperatures, 
the  amount  of  deviation  from  the  true  measure  is  dependent  upon 
the  extent  of  departure  from  such  fixed  location;  furthermore,  the 
nature  of  such  deviation,  in  a  positive  and  negative  sense,  is  depen- 
dent upon  the  direction  in  a  positive  and  negative  sense  of  such 
departure  on  the  temperature  scale. 

It  may  be  remarked  that  the  rate  of  such  deviation  from  the 
entropy  conception  is  not  the  same  for  all  substances. 

The  following  tabulation  shows  this  matter  plainly,  and  was 
derived  from  recently  published  tables  (Goodenough's)  for  satu- 
rated ammonia  (NH3)  which  are  believed  to  be  a  reliable  exposition 

dp 

of  experimentation  for  -5-  ,  "specific  volume  of  vapor,"  and  "speci- 
fic volume  of  liquid."  However,  in  deriving  the  values  for  "latent 
heat  of  evaporation,"  the  pernicious  practice  of  dispensing  with 
experimentation  and  co-ordinating  the  second  law  of  thermody- 
namics with  these  experimentally  derived  variables  was  adopted  in 
order  that  such  tables  would  be  compatible  with  the  so-called 
thermodynamic  laws;  consequently,  Goodenough's  tabulated 
values  for  "latent  heat  of  evaporation"  are  rejected  for  use  in  this 
investigation,  and  the  Dieterici  curve  is  accepted  as  a  substitute 
because  it  agrees  quite  nearly  with  experimentally  derived  values : 

At—  40    degrees  Fahr.,  deviation  from  entropy  =  —  15      per  cent 

tt  _j_  70          tt          n  tt          tt          tt       _      Q       tt      tt 

"+160  "          "  "       =  +  9-fr    "      " 

"+273.2      "          "    critical  point 

In  conclusion,  it  may  be  remarked  that  thorough  experimenta- 
tion, carried  into  extreme  regions  of  pressure  and  density,  will  con- 
tribute much  to  advance  the  science  of  motive  power  of  heat  and  the 
art  of  mechanical  refrigeration. 


